# Module 1 “Test Your Knowledge” Problem 1: ```Module 1
Problem 1: Nolan Ryan is traveling on a train traveling at 30 mph in the +x-direction
toward a batter. What is the speed of a Nolan Ryan fastball as seen by the batter if Nolan
Ryan can throw a fastball at 100 mph?
Problem 2: Darth Vader is traveling at 0.3 c toward Luke Sky Walker. Darth Vader fires
a pulse of light at Sky Walker.
A. According to classical mechanics, what is the speed of the light pulse seen by Luke
Sky Walker?
B.
According to electromagnetism, what is the speed of the light pulse seen by Luke
Sky Walker?
Problem 3: Show that the electromagnetic wave equation is not invariant under the
Galilean Transformation.
Problem 4: Approximate the following functions using the binomial approximation:
A.
1
where x 1
1 x
B.
1
where r  R
R r
C.
mo c 2
v
1  
c
2
 mo c 2 where v  c
Problem 5: Use the Taylor Series expansion to obtain the following small angle
approximation from Trigonometry:
A)
cos θ   1
B)
sin θ   θ
C)
Using dimensional analysis and the Taylor series explain why these
approximations are only valid for angles measured in radians.
Problem 6: Use the Taylor series expansions for sin(x), Cos(x), and ex to prove that
ei θ  cosθ   i sin θ .
i
π
Problem 7: Given the complex numbers z  3  i 5 , y  4  i 2 , q  3 e 4 , and
m  2e
i
5π
4
, solve for the following
A)
z+y
B)
q/m
C)
q3
```